Background for slopes:
#color(white)("XXXX")#The slope of a line is defined as
#color(white)("XXXX")##color(white)("XXXX")##(Delta y)/(Delta x)#
#color(white)("XXXX")#That is, given two points #(x_1,y_1)# and #(x_2,y_2)# on the line
#color(white)("XXXX")#the slope is
#color(white)("XXXX")##color(white)("XXXX")##m=(y_2-y_1)/(x_2-x_1)#
#color(white)("XXXX")#For a straight line the slope is the same for all pairs of points on the line
#color(white)("XXXX")#Therefore, given two fixed points (as above) and a variable point #(x,y)# on the line
#color(white)("XXXX")##color(white)("XXXX")##(y-y_1)/(x-x_1) = (y_2-y_1)/(x_2-x_1)#
#color(white)("XXXX")#This can be rewritten:
#color(white)("XXXX")##color(white)("XXXX")##y=m(x-x_1)+y_1#
#color(white)("XXXX")#If a line has a slope of #hatm# then all lines perpendicular to it have a slope of #1/(hatm)#
Slope of #6x-7y=6#
#color(white)("XXXX")#This equation can be rewritten as
#color(white)("XXXX")##color(white)("XXXX")##y = (6/7)x+(6/7)#
#color(white)("XXXX")#and therefore has a slope of #(6/7)#
#color(white)("XXXX")#Any line perpendicular to it has a slope of #(-7/6)#
Equation of a line through #(2,3)# perpendicular to #6x-7y=6#
#color(white)("XXXX")#Using the previous discussion:
#color(white)("XXXX")##color(white)("XXXX")##y = (-7/6)(x-2)+3#
#color(white)("XXXX")#or, simplified and re-written in function notation
#color(white)("XXXX")##color(white)("XXXX")##f(x) = -7/6x+2/3#
